The exactness of mathematics and the inexactness of physics. This post was prompted by something that came up in a previous one (i.e. see my blog on 12 August 2021), where I commented on the fact that an anonymous referee did not know what to make of an asymptotic curve. The obvious conclusion from thisContinue reading The exactness of mathematics and the inexactness of physics.

Nightmare on Buccleuch Street. Staycation post No 4. I will be out of the virtual office until 30 August. I haven’t been into the university since the pandemic began but recently I dreamt that I was in the university library, in the section where magazines and journals are kept. In this dream, I was sittingContinue reading Nightmare on Buccleuch Street.

Why am I so concerned about Onsager’s so-called conjecture? Staycation post No 3. I will be out of the virtual office until 30 August. In recent years, Onsager’s (1949) paper on turbulence has been rediscovered and its eccentricities promoted enthusiastically, despite the fact that they are at odds with much well-established research in turbulence, beginningContinue reading Why am I so concerned about Onsager’s so-called conjecture?

That’s the giddy limit! Staycation post No 2. I will be out of the virtual office until 30 August. The expression above was still in use when I was young, and vestiges of its use linger on even today. It referred, often jocularly, to any behaviour which was deemed unacceptable. Why giddy? I’m afraid thatContinue reading That’s the giddy limit!

When is a conjecture not a conjecture? Staycation post No 1. I will be out of the virtual office until 30 August. That sounds like the sort of riddle I used to hear in childhood. For instance, when is a door not a door? The answer was: when it’s ajar! [1] Well, at least weContinue reading When is a conjecture not a conjecture?

How do we identify the presence of turbulence? In 1971, when I began as a lecturer in Engineering Science at Edinburgh, my degree in physics provided me with no basis for teaching fluid dynamics. I had met the concept of the convective derivative in statistical mechanics, as part of the derivation of the Liouville equation,Continue reading How do we identify the presence of turbulence?

Are Kraichnan’s papers difficult to read? Part 2: The DIA. In 2008, or thereabouts, I took part in a small conference at the Isaac Newton Institute and gave a talk on the LET theory, its relationship to DIA, and how both theories could be understood in terms of their relationship to Quasi-normality. During my talk,Continue reading Are Kraichnan’s papers difficult to read? Part 2: The DIA.

Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance. When I was first at Edinburgh, in the early 1970s, I gave some informal talks on turbulence theory. One of my colleagues became sufficiently interested to start doing some reading on the subject. Shortly afterwards he came up to me at coffee time and said.Continue reading Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance

Hurrah for arXiv.com! In my previous blog, I referred to my paper with Michael May [1], which failed to be accepted for publication, despite my having tried several journals. I suppose that some of my choices were unrealistic (e.g Nature) and that I could have tried more. Also, I could have specified referees, which IContinue reading Hurrah for arXiv.com!

The Kolmogorov (1962) theory: a critical review Part 2 Following on to last week’s post, I would like to make a point that, so far as I know, has not previously been made in the literature of the subject. This is, that the energy spectrum is (in the sense of thermodynamics) an intensive quantity. ThereforeContinue reading The Kolmogorov (1962) theory: a critical review Part 2

The Kolmogorov (1962) theory: a critical review Part 1 As is well known, Kolmogorov interpreted Landau’s criticism as referring to the small-scale intermittency of the instantaneous dissipation rate. His response was to adopt Obukhov’s proposal to introduce a new dissipation rate which had been averaged over a sphere of radius , and which may beContinue reading The Kolmogorov (1962) theory: a critical review Part 1

The Landau criticism of K41 and problems with averages The idea that K41 had some problem with the way that averages were taken has its origins in the famous footnote on page 126 of the book by Landau and Lifshitz [1]. This footnote is notoriously difficult to understand; not least because it is meaningless unlessContinue reading The Landau criticism of K41 and problems with averages

The Kolmogorov-Obukhov Spectrum. To lay a foundation for the present piece, we will first consider the joint Kolmogorov-Obukhov picture in more detail. For completeness, we should begin by mentioning that Kolmogorov also used the Karman-Howarth equation, which is the energy balance equation connecting the second- and third-order structure functions, to derive the so-called `‘ lawContinue reading The Kolmogorov-Obukhov Spectrum.

Why do we call it ‘The Kolmogorov Spectrum’? The Kolmogorov spectrum continues to be the subject of contentious debate. Despite its great utility in applications and its overwhelming confirmation by experiments, it is still plagued by the idea that it is subject to intermittency corrections. From a fundamental view this is difficult to understand becauseContinue reading Why do we call it ‘The Kolmogorov Spectrum’?

The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory. In last week’s blog, I discussed the Kraichnan and Wyld approaches to the turbulence closure problem. These field-theoretic approaches are examples of RPTs, while the pioneering theory of Edwards [1] is a self-consistent field theory. An interesting differenceContinue reading The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory.

What if anything is wrong with Wyld’s (1962) turbulence formulation? When I began my PhD in 1966, I found Wyld’s paper [1] to be one of the easiest to understand. However, one feature of the formalism struck me as odd or incorrect, so I didn’t spend any more time on it. But I had foundContinue reading What if anything is wrong with Wyld’s (1962) turbulence formulation?

Is turbulence research still in its infancy? Recently I came across the article by Lumley and Yaglom which is cited below as [1]. I think it is new to me but quite possibly I will find it lurking in my filing system when at last I am able to visit my university office again. ItContinue reading Is turbulence research still in its infancy?

Culture wars: applied scientists versus natural scientists. In my early years at Edinburgh, I attended a seminar on polymer drag reduction; and, as I was walking back with a small group, we were discussing what we had just learned. In response to a comment made by one member of the group, I observed that itContinue reading Culture wars: applied scientists versus natural scientists.

‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744) I have written about the problems posed by the different cultures to be found in the turbulence community; and in particular of the difficulties faced by some referees when confronted by Fourier methods. My interest in the matter is of course the difficulties faced byContinue reading ‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744)

Intermittency, intermittency, intermittency! It is well known that those who are concerned with the sale of property say that the three factors determining the value of a house are: location, location, location. In fact I believe that there is a television programme with that as a title. This trope has passed into the general consciousness;Continue reading Intermittency, intermittency, intermittency!

Does the failure to use spectral methods harm one’s understanding of turbulence? Vacation post No. 3: I will be out of the virtual office until Monday 19 April. As described in the previous post, traditional methods of visualising turbulence involve vaguely specified and ill-defined eddying motions whereas Fourier methods lead to a well-defined problem inContinue reading Does the failure to use spectral methods harm one’s understanding of turbulence?

Does the use of spectral methods obscure the physics of turbulence? Vacation post No. 2: I will be out of the virtual office until Monday 19 April. Recently, someone who posted a comment on one of my early blogs about spectral methods (see the post on 20 February 2020), commented that a certain person hasContinue reading Does the use of spectral methods obscure the physics of turbulence?

Stirring forces and the turbulence response. Vacation post No. I: I will be out of the office until Monday 19 April. In my previous post, I argued that there seems to be really no justification for regarding the stirring forces that we invoke in isotropic turbulence as mysterious, at least in the context of statisticalContinue reading Stirring forces and the turbulence response.

The mysterious stirring forces In the late 1970s there was an upsurge in interest in the turbulence problem among theoretical physicists. This arose out of the application of renormalization group (RG) methods to the problem of stirred fluid motion. As this problem was restricted to a very low wavenumber cutoff, these approaches had nothing toContinue reading The mysterious stirring forces

Is the entropy of turbulence a maximum? In 1969 I published my first paper [1], jointly with my supervisor Sam Edwards, in which we maximised the turbulent entropy, defined in terms of the information content, in order to obtain a prescription for , the renormalized decay time for the energy contained in the mode withContinue reading Is the entropy of turbulence a maximum?

Analogies between critical phenomena and turbulence: 2 In the previous post, I discussed the misapplication to turbulence of concepts like the relationship between mean-field theory and Renormalization Group in critical phenomena. This week I have the concept of ‘anomalous exponents’ in my sights! This term appears to be borrowed from the concept of anomalous dimensionContinue reading Analogies between critical phenomena and turbulence: 2

Analogies between critical phenomena and turbulence: 1 In the late 1970s, application of Renormalization Group (RG) to stirred fluid motion led to an upwelling of interest among theoretical physicists in the possibility of solving the notorious turbulence problem. I remember reading a conference paper which included some discussion that was rather naïve in tone. ForContinue reading Analogies between critical phenomena and turbulence: 1

Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis. Earlier this year I received an enquiry from Alex Liberzon, who was puzzled by the fact that some people plot temporal frequency spectra with a power law, but he was unable to reconcile the dimensions. This immediately took me back to the 1970s when IContinue reading Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis.

The concept of universality classes in critical phenomena. The universality of the small scales, which is predicted by the Richardson-Kolmogorov picture, is not always observed in practice; and in the previous post I conjectured that departures from this might be accounted for by differences in the spatial symmetry of the large scale flow. To takeContinue reading The concept of universality classes in critical phenomena.

Macroscopic symmetry and microscopic universality. The concepts of macroscopic and microscopic are often borrowed, in an unacknowledged way, from physics, in order to think about the fundamentals of turbulence. By that, I mean that there is usually no explicit acknowledgement, nor indeed apparent realization, that the ratio of large scales to small scales is manyContinue reading Macroscopic symmetry and microscopic universality.

Can statistical theory help with turbulence modelling? When reading the book by Sagaut and Cambon some years ago, I was struck by their balance between fundamentals and applications [1]. This started me thinking, and it appeared to me that I had become ever more concentrated on fundamentals in recent years. In other words, I seemedContinue reading Can statistical theory help with turbulence modelling?

The last post … of the first year! A year ago, when I began this blog, few of us can have had any idea of what the year had in store from the coronavirus, now known to us as covid-19. Over the years, I have sometimes reflected on the very fortunate lives of my generation.Continue reading The last post … of the first year!

How important are the higher-order moments of the velocity field? Up until about 1970, fundamental work on turbulence was dominated by the study of the energy spectrum, and most work was carried out in wavenumber space. In 1963 Uberoi measured the time-derivative of the energy spectrum and also the dissipation spectrum, in grid turbulence; andContinue reading How important are the higher-order moments of the velocity field?

How big is infinity? In physics it is usual to derive theories of macroscopic systems by taking an infinite limit. This could be the continuum limit or the thermodynamic limit. Or, in the theory of critical phenomena, the signal of a nontrivial fixed point is that the correlation length becomes infinite. Of course, what weContinue reading How big is infinity?

My life in wavenumber space In September 1966, when I began work on my PhD, I almost immediately began to dwell in wavenumber space. After a brief nod to the real-space equations, I had to learn about Fourier transformation of the velocity field, with the wave-vector replacing the position vector , and the Navier-Stokes equationsContinue reading My life in wavenumber space

How many angels can dance on the point of a pin? When I was young this was often quoted as an example of the foolishness of the medieval schoolmen and the nonsensical nature of their discussions. I happily classed those who debated it along with those who, not only believed that the sun was pulledContinue reading How many angels can dance on the point of a pin?

Academic fathers and Mother Christmas In the mid-1980s I visited the Max Planck institute in Bonn to give a talk. While I was there, some of the German mathematicians told me about the concept of an academic father. They said that your PhD supervisor was your academic father, his supervisor was your academic grandfather, andContinue reading Academic fathers and Mother Christmas

Peer Review: Through the Looking Glass Five years ago, when carrying out direct numerical simulations (DNS) of isotropic turbulence at Edinburgh, we made a surprising discovery. We found that turbulence states died away at very low values of the Reynolds number and the flow became self-organised, taking the form of a Beltrami flow, which hasContinue reading Peer Review: Through the Looking Glass

Should theories of turbulence be intelligible to fluid dynamicists? One half of the Nobel Prize in physics for 2020 was awarded to Roger Penrose for demonstrating that ‘black hole formation is a robust prediction of the General Theory of Relativity’. While it’s not my field, I do know a little about general relativity; so IContinue reading Should theories of turbulence be intelligible to fluid dynamicists?

Turbulent dissipation and the two cultures? I recently saw the paper cited as [1] below, which for me is I think the first of the 2021 papers. As the title suggests, it presents a review of methods of measuring the turbulent dissipation rate. It contains a certain amount of basic theory, along the lines ofContinue reading Turbulent dissipation and the two cultures?

The infinite Reynolds number limit: Onsager versus Batchelor: 3 In the preceding two posts, we have pointed out that the final statement by Onsager in his 1949 paper [1] is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor [2] and extendedContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 3

The infinite Reynolds number limit: Onsager versus Batchelor: 2 In the preceding post, we argued that the final statement by Onsager in his 1949 paper [1] is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor [2] and extended by Edwards [3],Continue reading The infinite Reynolds number limit: Onsager versus Batchelor: 2

The infinite Reynolds number limit: Onsager versus Batchelor: 1 A pioneering paper on turbulence by Onsager, which was published in 1949 [1], seems to have had a profound influence on some aspects of the subject in later years. In particular, he put forward the idea that as the turbulence was still dissipative in the limitContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 1

The role of Gaussians in turbulence studies. The Gaussian, or normal, distribution plays a key part in statistical field theory. This is partly because it is the only functional which can be integrated and partly because Gaussian distributions are frequently encountered in microscopic physics at, or near, thermal equilibrium. The latter is not the caseContinue reading The role of Gaussians in turbulence studies.

Is there actually a single ‘turbulence problem’? When I was preparing last week’s post, I consulted the Saffman lectures in order to find an example of the culture clash between theoretical physics and applied maths. In the process I noticed quite a few points that I felt tempted to write about and in particular thatContinue reading Is there actually a single ‘turbulence problem’?

Here’s to mathematics and may it never be of use to anyone! When I was a student, I read that mathematicians at conference dinners would drink a toast along the lines of the title of this piece. As an idealistic young man, I was quite shocked by this; and thought it very arrogant. Apart fromContinue reading Here’s to mathematics and may it never be of use to anyone!

Operational Large-Eddy Simulation. When I was visiting TU Delft in 1997, I stayed with my wife and daughter in the Hague, where we rented an apartment from one of the professors at Delft. He and his wife occupied the penthouse above us. They had originally bought the second apartment so that their teenage daughters couldContinue reading Operational Large-Eddy Simulation.

Peer Review in Wonderland In 1974 I completed a task which had begun during my PhD days, and found a way of rendering the Edwards statistical closure compatible with the Kolmogorov spectrum. The basic idea was that the entire transfer spectrum acted as a sink of energy at low wavenumbers and a source of energyContinue reading Peer Review in Wonderland

Formulation of Renormalization Group (RG) for turbulence: 2 In last week’s post, we recognised that the basic step of averaging over high-frequency modes was impossible in principle for a classical, deterministic problem such as turbulence. Curiously enough, for many years it has been recognized in the analogous subgrid modelling problem that a conditional average isContinue reading Formulation of Renormalization Group (RG) for turbulence: 2

Formulation of Renormalization Group (RG) for turbulence: 1 In my posts of 30 April and 7 May, I discussed the relevance of field-theoretic methods (and particularly RG) to the Navier-Stokes equation (NSE). Here I want to deal with some specific points and in the process highlight the snags involved in going from microscopic quantum randomnessContinue reading Formulation of Renormalization Group (RG) for turbulence: 1

Reynolds averaging re-formulated. At the beginning of the 1980s I was still involved in experimental work on drag reduction; while, on the theoretical side, I had begun numerical evaluation of the LET theory. One day I went into the lab to help a student who was having problems with his laser anemometer. In those daysContinue reading Reynolds averaging re-formulated.

Turbulent dissipation and other rates of change. When I was working for my PhD with Sam Edwards in the late 1960s, my second supervisor was David Leslie. We would meet up every so often to discuss progress, and I recall that David was invariably exasperated by our concentration on asymptotic behaviour at high wavenumbers. HeContinue reading Turbulent dissipation and other rates of change.